<!DOCTYPE html>
<html lang="zh-CN">
<head>
  <meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=2">
<meta name="theme-color" content="#222">
<meta name="generator" content="Hexo 5.2.0">


  <link rel="apple-touch-icon" sizes="180x180" href="/images/apple-touch-icon-next.png">
  <link rel="icon" type="image/png" sizes="32x32" href="/images/favicon-32x32-next.png">
  <link rel="icon" type="image/png" sizes="16x16" href="/images/favicon-16x16-next.png">
  <link rel="mask-icon" href="/images/logo.svg" color="#222">
  <meta name="google-site-verification" content="gKNVOYbFPuBJzjqvnn652-CYWF2U-lQiejoLTXhZfwk">
  <meta name="msvalidate.01" content="2D8978DCD4204A3D25D21A1EDB8E1FD8">
  <meta name="baidu-site-verification" content="code-w1HF6XYOAb">

<link rel="stylesheet" href="/css/main.css">



<link rel="stylesheet" href="//cdn.jsdelivr.net/npm/@fortawesome/fontawesome-free@5.15.1/css/all.min.css">
  <link rel="stylesheet" href="//cdn.jsdelivr.net/npm/animate.css@3.1.1/animate.min.css">

<script class="hexo-configurations">
    var NexT = window.NexT || {};
    var CONFIG = {"hostname":"ysl970629.github.io","root":"/","images":"/images","scheme":"Gemini","version":"8.0.2","exturl":false,"sidebar":{"position":"left","display":"post","padding":18,"offset":12},"copycode":false,"bookmark":{"enable":true,"color":"#222","save":"auto"},"fancybox":false,"mediumzoom":false,"lazyload":false,"pangu":false,"comments":{"style":"tabs","active":"valine","storage":true,"lazyload":false,"nav":null,"activeClass":"valine"},"motion":{"enable":true,"async":false,"transition":{"post_block":"fadeIn","post_header":"fadeInDown","post_body":"fadeInDown","coll_header":"fadeInLeft","sidebar":"fadeInUp"}},"prism":false,"i18n":{"placeholder":"搜索...","empty":"没有找到任何搜索结果：${query}","hits_time":"找到 ${hits} 个搜索结果（用时 ${time} 毫秒）","hits":"找到 ${hits} 个搜索结果"},"path":"/search.xml","localsearch":{"enable":true,"trigger":"auto","top_n_per_article":1,"unescape":false,"preload":false}};
  </script>
<meta name="description" content="由于微分方程有很多种类，在做题时需要快速判别。因此我整理了一下分类，有助于自己理解。 只是伪代码表示，如果直接在电脑上运行，肯定是没法运行的。">
<meta property="og:type" content="article">
<meta property="og:title" content="判断微分方程的类型（Java伪代码表示）">
<meta property="og:url" content="https://ysl970629.github.io/posts/d6882305/index.html">
<meta property="og:site_name" content="YuSLi的部落阁">
<meta property="og:description" content="由于微分方程有很多种类，在做题时需要快速判别。因此我整理了一下分类，有助于自己理解。 只是伪代码表示，如果直接在电脑上运行，肯定是没法运行的。">
<meta property="og:locale" content="zh_CN">
<meta property="article:published_time" content="2020-05-07T03:38:37.000Z">
<meta property="article:modified_time" content="2020-12-02T09:20:30.317Z">
<meta property="article:author" content="YuSLi">
<meta property="article:tag" content="数学二">
<meta property="article:tag" content="高数">
<meta property="article:tag" content="微分方程">
<meta name="twitter:card" content="summary">


<link rel="canonical" href="https://ysl970629.github.io/posts/d6882305/">


<script class="page-configurations">
  // https://hexo.io/docs/variables.html
  CONFIG.page = {
    sidebar: "",
    isHome : false,
    isPost : true,
    lang   : 'zh-CN'
  };
</script>
<title>判断微分方程的类型（Java伪代码表示） | YuSLi的部落阁</title>
  



  <noscript>
  <style>
  body { margin-top: 2rem; }

  .use-motion .menu-item,
  .use-motion .sidebar,
  .use-motion .post-block,
  .use-motion .pagination,
  .use-motion .comments,
  .use-motion .post-header,
  .use-motion .post-body,
  .use-motion .collection-header {
    visibility: visible;
  }

  .use-motion .header,
  .use-motion .site-brand-container .toggle,
  .use-motion .footer { opacity: initial; }

  .use-motion .site-title,
  .use-motion .site-subtitle,
  .use-motion .custom-logo-image {
    opacity: initial;
    top: initial;
  }

  .use-motion .logo-line {
    transform: scaleX(1);
  }

  .search-pop-overlay, .sidebar-nav { display: none; }
  .sidebar-panel { display: block; }
  </style>
</noscript>

</head>

<body itemscope itemtype="http://schema.org/WebPage" class="use-motion">
  <div class="headband"></div>

  <main class="main">
    <header class="header" itemscope itemtype="http://schema.org/WPHeader">
      <div class="header-inner"><div class="site-brand-container">
  <div class="site-nav-toggle">
    <div class="toggle" aria-label="切换导航栏">
        <span class="toggle-line"></span>
        <span class="toggle-line"></span>
        <span class="toggle-line"></span>
    </div>
  </div>

  <div class="site-meta">

    <a href="/" class="brand" rel="start">
      <i class="logo-line"></i>
      <h1 class="site-title">YuSLi的部落阁</h1>
      <i class="logo-line"></i>
    </a>
      <p class="site-subtitle" itemprop="description">默默无闻的小卒</p>
  </div>

  <div class="site-nav-right">
    <div class="toggle popup-trigger">
        <i class="fa fa-search fa-fw fa-lg"></i>
    </div>
  </div>
</div>



<nav class="site-nav">
  <ul class="main-menu menu">
        <li class="menu-item menu-item-home">

    <a href="/" rel="section"><i class="fa fa-home fa-fw"></i>首页</a>

  </li>
        <li class="menu-item menu-item-tags">

    <a href="/tags/" rel="section"><i class="fa fa-tags fa-fw"></i>标签</a>

  </li>
        <li class="menu-item menu-item-categories">

    <a href="/categories/" rel="section"><i class="fa fa-th fa-fw"></i>分类</a>

  </li>
        <li class="menu-item menu-item-archives">

    <a href="/archives/" rel="section"><i class="fa fa-archive fa-fw"></i>归档</a>

  </li>
        <li class="menu-item menu-item-about">

    <a href="/about/" rel="section"><i class="fa fa-user fa-fw"></i>关于</a>

  </li>
        <li class="menu-item menu-item-schedule">

    <a href="/schedule/" rel="section"><i class="fa fa-calendar fa-fw"></i>日程表</a>

  </li>
        <li class="menu-item menu-item-sitemap">

    <a href="/sitemap.xml" rel="section"><i class="fa fa-sitemap fa-fw"></i>站点地图</a>

  </li>
        <li class="menu-item menu-item-commonweal">

    <a href="/404/" rel="section"><i class="fa fa-heartbeat fa-fw"></i>公益 404</a>

  </li>
      <li class="menu-item menu-item-search">
        <a role="button" class="popup-trigger"><i class="fa fa-search fa-fw"></i>搜索
        </a>
      </li>
  </ul>
</nav>



  <div class="search-pop-overlay">
    <div class="popup search-popup"><div class="search-header">
  <span class="search-icon">
    <i class="fa fa-search"></i>
  </span>
  <div class="search-input-container">
    <input autocomplete="off" autocapitalize="off" maxlength="80"
           placeholder="搜索..." spellcheck="false"
           type="search" class="search-input">
  </div>
  <span class="popup-btn-close">
    <i class="fa fa-times-circle"></i>
  </span>
</div>
<div class="search-result-container no-result">
  <div class="search-result-icon">
    <i class="fa fa-spinner fa-pulse fa-5x"></i>
  </div>
</div>

    </div>
  </div>

</div>
        
  
  <div class="toggle sidebar-toggle">
    <span class="toggle-line"></span>
    <span class="toggle-line"></span>
    <span class="toggle-line"></span>
  </div>

  <aside class="sidebar">

    <div class="sidebar-inner sidebar-overview-active">
      <ul class="sidebar-nav">
        <li class="sidebar-nav-toc">
          文章目录
        </li>
        <li class="sidebar-nav-overview">
          站点概览
        </li>
      </ul>

      <div class="sidebar-panel-container">
        <!--noindex-->
        <section class="post-toc-wrap sidebar-panel">
        </section>
        <!--/noindex-->

        <section class="site-overview-wrap sidebar-panel">
          <div class="site-author site-overview-item animated" itemprop="author" itemscope itemtype="http://schema.org/Person">
    <img class="site-author-image" itemprop="image" alt="YuSLi"
      src="https://cdn.jsdelivr.net/gh/ysl970629/public_picture_bed_01@latest//img/00头像.jpg#/images/avatar.gif">
  <p class="site-author-name" itemprop="name">YuSLi</p>
  <div class="site-description" itemprop="description"></div>
</div>
<div class="site-state-wrap site-overview-item animated">
  <nav class="site-state">
      <div class="site-state-item site-state-posts">
          <a href="/archives/">
        
          <span class="site-state-item-count">10</span>
          <span class="site-state-item-name">日志</span>
        </a>
      </div>
      <div class="site-state-item site-state-categories">
            <a href="/categories/">
          
        <span class="site-state-item-count">5</span>
        <span class="site-state-item-name">分类</span></a>
      </div>
      <div class="site-state-item site-state-tags">
            <a href="/tags/">
          
        <span class="site-state-item-count">10</span>
        <span class="site-state-item-name">标签</span></a>
      </div>
  </nav>
</div>
  <div class="links-of-author site-overview-item animated">
      <span class="links-of-author-item">
        <a href="https://github.com/ysl970629" title="GitHub → https:&#x2F;&#x2F;github.com&#x2F;ysl970629" rel="noopener" target="_blank"><i class="fab fa-github fa-fw"></i>GitHub</a>
      </span>
      <span class="links-of-author-item">
        <a href="mailto:ysl_0629@qq.com" title="E-Mail → mailto:ysl_0629@qq.com" rel="noopener" target="_blank"><i class="fa fa-envelope fa-fw"></i>E-Mail</a>
      </span>
      <span class="links-of-author-item">
        <a href="https://weibo.com/u/6003135640?is_all=1" title="Weibo → https:&#x2F;&#x2F;weibo.com&#x2F;u&#x2F;6003135640?is_all&#x3D;1" rel="noopener" target="_blank"><i class="fab fa-weibo fa-fw"></i>Weibo</a>
      </span>
      <span class="links-of-author-item">
        <a href="https://twitter.com/yusli0629_" title="Twitter → https:&#x2F;&#x2F;twitter.com&#x2F;yusli0629_" rel="noopener" target="_blank"><i class="fab fa-twitter fa-fw"></i>Twitter</a>
      </span>
      <span class="links-of-author-item">
        <a href="https://www.facebook.com/profile.php?id=100039363857543" title="FB Page → https:&#x2F;&#x2F;www.facebook.com&#x2F;profile.php?id&#x3D;100039363857543" rel="noopener" target="_blank"><i class="fab fa-facebook fa-fw"></i>FB Page</a>
      </span>
      <span class="links-of-author-item">
        <a href="https://stackoverflow.com/users/13379393" title="StackOverflow → https:&#x2F;&#x2F;stackoverflow.com&#x2F;users&#x2F;13379393" rel="noopener" target="_blank"><i class="fab fa-stack-overflow fa-fw"></i>StackOverflow</a>
      </span>
      <span class="links-of-author-item">
        <a href="https://www.instagram.com/ysl0629_/" title="Instagram → https:&#x2F;&#x2F;www.instagram.com&#x2F;ysl0629_&#x2F;" rel="noopener" target="_blank"><i class="fab fa-instagram fa-fw"></i>Instagram</a>
      </span>
      <span class="links-of-author-item">
        <a href="https://space.bilibili.com/8531521/" title="Bilibili → https:&#x2F;&#x2F;space.bilibili.com&#x2F;8531521&#x2F;" rel="noopener" target="_blank"><i class="fab fa-bilibili fa-fw"></i>Bilibili</a>
      </span>
  </div>
  <div class="cc-license site-overview-item animated" itemprop="license">
    <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" class="cc-opacity" rel="noopener" target="_blank"><img src="/images/cc-by-nc-sa.svg" alt="Creative Commons"></a>
  </div>


  <div class="links-of-blogroll site-overview-item animated">
    <div class="links-of-blogroll-title"><i class="fa fa-globe fa-fw"></i>
      Links
    </div>
    <ul class="links-of-blogroll-list">
        <li class="links-of-blogroll-item">
          <a href="https://theme-next.js.org/" title="https:&#x2F;&#x2F;theme-next.js.org&#x2F;" rel="noopener" target="_blank">hexo-theme-next博客主页</a>
        </li>
        <li class="links-of-blogroll-item">
          <a href="https://ysl970629.github.io/" title="https:&#x2F;&#x2F;ysl970629.github.io&#x2F;">本站国外地址(Github)</a>
        </li>
        <li class="links-of-blogroll-item">
          <a href="https://ysl970629.gitee.io/" title="https:&#x2F;&#x2F;ysl970629.gitee.io&#x2F;" rel="noopener" target="_blank">本站国内地址(Gitee)</a>
        </li>
        <li class="links-of-blogroll-item">
          <a href="https://ajq7rq.coding-pages.com/" title="https:&#x2F;&#x2F;ajq7rq.coding-pages.com&#x2F;" rel="noopener" target="_blank">本站国内地址(Coding)</a>
        </li>
        <li class="links-of-blogroll-item">
          <a href="https://github.com/ysl970629/my-notes/" title="https:&#x2F;&#x2F;github.com&#x2F;ysl970629&#x2F;my-notes&#x2F;" rel="noopener" target="_blank">个人知识库</a>
        </li>
    </ul>
  </div>
<div class="cc-license animated" itemprop="sponsor">
  <a href="https://www.netlify.com" class="cc-opacity" title="Deploy with Netlify → https://www.netlify.com" target="_blank"><img width="80" src="https://www.netlify.com/img/global/badges/netlify-dark.svg" alt="Netlify"></a>
</div>

        </section>
      </div>
        <div class="back-to-top animated">
          <i class="fa fa-arrow-up"></i>
          <span>0%</span>
        </div>
    </div>
  </aside>
  <div class="sidebar-dimmer"></div>


    </header>

    
  <a role="button" class="book-mark-link book-mark-link-fixed"></a>

  <a href="https://github.com/ysl970629" class="github-corner" title="Follow me on GitHub" aria-label="Follow me on GitHub" rel="noopener" target="_blank"><svg width="80" height="80" viewBox="0 0 250 250" aria-hidden="true"><path d="M0,0 L115,115 L130,115 L142,142 L250,250 L250,0 Z"></path><path d="M128.3,109.0 C113.8,99.7 119.0,89.6 119.0,89.6 C122.0,82.7 120.5,78.6 120.5,78.6 C119.2,72.0 123.4,76.3 123.4,76.3 C127.3,80.9 125.5,87.3 125.5,87.3 C122.9,97.6 130.6,101.9 134.4,103.2" fill="currentColor" style="transform-origin: 130px 106px;" class="octo-arm"></path><path d="M115.0,115.0 C114.9,115.1 118.7,116.5 119.8,115.4 L133.7,101.6 C136.9,99.2 139.9,98.4 142.2,98.6 C133.8,88.0 127.5,74.4 143.8,58.0 C148.5,53.4 154.0,51.2 159.7,51.0 C160.3,49.4 163.2,43.6 171.4,40.1 C171.4,40.1 176.1,42.5 178.8,56.2 C183.1,58.6 187.2,61.8 190.9,65.4 C194.5,69.0 197.7,73.2 200.1,77.6 C213.8,80.2 216.3,84.9 216.3,84.9 C212.7,93.1 206.9,96.0 205.4,96.6 C205.1,102.4 203.0,107.8 198.3,112.5 C181.9,128.9 168.3,122.5 157.7,114.1 C157.9,116.9 156.7,120.9 152.7,124.9 L141.0,136.5 C139.8,137.7 141.6,141.9 141.8,141.8 Z" fill="currentColor" class="octo-body"></path></svg></a>

<noscript>
  <div class="noscript-warning">Theme NexT works best with JavaScript enabled</div>
</noscript>


    <div class="main-inner post posts-expand">


  


<div class="post-block">
  
  

  <article itemscope itemtype="http://schema.org/Article" class="post-content" lang="zh-CN">
    <link itemprop="mainEntityOfPage" href="https://ysl970629.github.io/posts/d6882305/">

    <span hidden itemprop="author" itemscope itemtype="http://schema.org/Person">
      <meta itemprop="image" content="https://cdn.jsdelivr.net/gh/ysl970629/public_picture_bed_01@latest//img/00头像.jpg#/images/avatar.gif">
      <meta itemprop="name" content="YuSLi">
      <meta itemprop="description" content="">
    </span>

    <span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization">
      <meta itemprop="name" content="YuSLi的部落阁">
    </span>
      <header class="post-header">
        <h1 class="post-title" itemprop="name headline">
          判断微分方程的类型（Java伪代码表示）
        </h1>

        <div class="post-meta-container">
          <div class="post-meta">
    <span class="post-meta-item">
      <span class="post-meta-item-icon">
        <i class="far fa-calendar"></i>
      </span>
      <span class="post-meta-item-text">发表于</span>

      <time title="创建时间：2020-05-07 11:38:37" itemprop="dateCreated datePublished" datetime="2020-05-07T11:38:37+08:00">2020-05-07</time>
    </span>
    <span class="post-meta-item">
      <span class="post-meta-item-icon">
        <i class="far fa-folder"></i>
      </span>
      <span class="post-meta-item-text">分类于</span>
        <span itemprop="about" itemscope itemtype="http://schema.org/Thing">
          <a href="/categories/%E8%80%83%E7%A0%94/" itemprop="url" rel="index"><span itemprop="name">考研</span></a>
        </span>
    </span>

  
  
  <span class="post-meta-item">
    
      <span class="post-meta-item-icon">
        <i class="far fa-comment"></i>
      </span>
      <span class="post-meta-item-text">Valine：</span>
    
    <a title="valine" href="/posts/d6882305/#valine-comments" itemprop="discussionUrl">
      <span class="post-comments-count valine-comment-count" data-xid="/posts/d6882305/" itemprop="commentCount"></span>
    </a>
  </span>
  
  
      </div>
      <div class="post-meta">
    <span class="post-meta-item" title="本文字数">
      <span class="post-meta-item-icon">
        <i class="far fa-file-word"></i>
      </span>
      <span class="post-meta-item-text">本文字数：</span>
      <span>6.4k</span>
    </span>
    <span class="post-meta-item" title="阅读时长">
      <span class="post-meta-item-icon">
        <i class="far fa-clock"></i>
      </span>
      <span class="post-meta-item-text">阅读时长 &asymp;</span>
      <span>11 分钟</span>
    </span>
</div>

        </div>
      </header>

    
    
    
    <div class="post-body" itemprop="articleBody">
        <p>由于微分方程有很多种类，在做题时需要快速判别。因此我整理了一下分类，有助于自己理解。</p>
<p>只是伪代码表示，如果直接在电脑上运行，肯定是没法运行的。</p>
<a id="more"></a>
<figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br><span class="line">101</span><br><span class="line">102</span><br><span class="line">103</span><br><span class="line">104</span><br><span class="line">105</span><br><span class="line">106</span><br><span class="line">107</span><br><span class="line">108</span><br><span class="line">109</span><br><span class="line">110</span><br><span class="line">111</span><br><span class="line">112</span><br><span class="line">113</span><br><span class="line">114</span><br><span class="line">115</span><br><span class="line">116</span><br><span class="line">117</span><br><span class="line">118</span><br><span class="line">119</span><br><span class="line">120</span><br><span class="line">121</span><br><span class="line">122</span><br><span class="line">123</span><br><span class="line">124</span><br><span class="line">125</span><br><span class="line">126</span><br><span class="line">127</span><br><span class="line">128</span><br><span class="line">129</span><br><span class="line">130</span><br><span class="line">131</span><br><span class="line">132</span><br><span class="line">133</span><br><span class="line">134</span><br><span class="line">135</span><br><span class="line">136</span><br><span class="line">137</span><br><span class="line">138</span><br><span class="line">139</span><br><span class="line">140</span><br><span class="line">141</span><br><span class="line">142</span><br><span class="line">143</span><br><span class="line">144</span><br><span class="line">145</span><br><span class="line">146</span><br><span class="line">147</span><br><span class="line">148</span><br><span class="line">149</span><br><span class="line">150</span><br><span class="line">151</span><br><span class="line">152</span><br><span class="line">153</span><br><span class="line">154</span><br><span class="line">155</span><br><span class="line">156</span><br><span class="line">157</span><br><span class="line">158</span><br><span class="line">159</span><br><span class="line">160</span><br><span class="line">161</span><br><span class="line">162</span><br><span class="line">163</span><br><span class="line">164</span><br><span class="line">165</span><br><span class="line">166</span><br><span class="line">167</span><br><span class="line">168</span><br><span class="line">169</span><br><span class="line">170</span><br><span class="line">171</span><br><span class="line">172</span><br><span class="line">173</span><br><span class="line">174</span><br><span class="line">175</span><br><span class="line">176</span><br><span class="line">177</span><br><span class="line">178</span><br><span class="line">179</span><br><span class="line">180</span><br><span class="line">181</span><br><span class="line">182</span><br><span class="line">183</span><br><span class="line">184</span><br><span class="line">185</span><br><span class="line">186</span><br><span class="line">187</span><br><span class="line">188</span><br><span class="line">189</span><br><span class="line">190</span><br><span class="line">191</span><br><span class="line">192</span><br><span class="line">193</span><br><span class="line">194</span><br><span class="line">195</span><br><span class="line">196</span><br><span class="line">197</span><br><span class="line">198</span><br><span class="line">199</span><br><span class="line">200</span><br><span class="line">201</span><br><span class="line">202</span><br><span class="line">203</span><br><span class="line">204</span><br><span class="line">205</span><br><span class="line">206</span><br><span class="line">207</span><br><span class="line">208</span><br><span class="line">209</span><br><span class="line">210</span><br><span class="line">211</span><br><span class="line">212</span><br><span class="line">213</span><br><span class="line">214</span><br><span class="line">215</span><br><span class="line">216</span><br><span class="line">217</span><br><span class="line">218</span><br><span class="line">219</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> 计算微分方程的解 </span>&#123;</span><br><span class="line">	<span class="keyword">private</span> 计算微分方程的解() &#123;&#125;</span><br><span class="line">	<span class="comment">/**</span></span><br><span class="line"><span class="comment">	 * 判断微分方程类型的方法，返回微分方程的类型</span></span><br><span class="line"><span class="comment">	 * </span></span><br><span class="line"><span class="comment">	 * <span class="doctag">@param</span> 方程: 待判断的微分方程</span></span><br><span class="line"><span class="comment">	 * <span class="doctag">@return</span> 微分方程的种类</span></span><br><span class="line"><span class="comment">	 */</span></span><br><span class="line">	<span class="keyword">public</span> <span class="keyword">static</span> String 判断微分方程类型(微分方程 方程) &#123;</span><br><span class="line">        <span class="comment">//对于一阶方程：</span></span><br><span class="line">		<span class="keyword">if</span> (方程.阶数 &lt;= <span class="number">1</span>) &#123;</span><br><span class="line">             <span class="comment">//第一种</span></span><br><span class="line">			<span class="keyword">if</span>(方程.is(<span class="string">&quot;可因式分解&quot;</span>)) &#123;</span><br><span class="line">				<span class="keyword">return</span> <span class="string">&quot;可分离变量型，分离变量后两边积分&quot;</span>;</span><br><span class="line">			&#125;</span><br><span class="line">             <span class="comment">//第二种</span></span><br><span class="line">			<span class="keyword">if</span> (方程.is(<span class="string">&quot;可化为φ(y/x)形式&quot;</span>)) &#123;</span><br><span class="line">				<span class="keyword">return</span> <span class="string">&quot;齐次型，带公式&quot;</span>;</span><br><span class="line">			&#125;</span><br><span class="line">             <span class="comment">//剩下的就都是线性的了；y&#x27; + b(x)*y = c(x)，包括齐次与非齐次两种</span></span><br><span class="line">             <span class="comment">//第三种</span></span><br><span class="line">			<span class="keyword">if</span> (方程.c(x).equals(<span class="string">&quot;0&quot;</span>)) &#123;</span><br><span class="line">				<span class="keyword">return</span> <span class="string">&quot;一阶线性齐次型，带公式&quot;</span>;</span><br><span class="line">			&#125;</span><br><span class="line">             <span class="comment">//第四种</span></span><br><span class="line">             <span class="comment">//去掉线性齐次后，剩下的都是线性非齐次。即Q(x) != 0</span></span><br><span class="line">             System.out.println(<span class="string">&quot;非齐次型方程的通解 = 对应的齐次的通解 + 非齐次的特解&quot;</span>);</span><br><span class="line">             <span class="comment">//上述是线性微分方程的解的“叠加原理”。不仅一阶，高阶的也具有此特点</span></span><br><span class="line">             <span class="keyword">return</span> <span class="string">&quot;一阶线性非齐次型，带公式&quot;</span>;</span><br><span class="line">		&#125;</span><br><span class="line">        <span class="comment">//除了一阶方程，剩下的都是高阶方程</span></span><br><span class="line">        </span><br><span class="line">        <span class="comment">//对于可降阶的高阶，有三种情况：</span></span><br><span class="line">        <span class="comment">//情况1</span></span><br><span class="line">        <span class="keyword">if</span>(方程.is(<span class="string">&quot;f(x, y^(n)) = 0 类型&quot;</span>)) &#123;</span><br><span class="line">            System.out.println(<span class="string">&quot;只有y^(n)，没有y&#x27;和y&quot;</span>);</span><br><span class="line">            <span class="keyword">return</span> <span class="string">&quot;第一种可降阶，连续积分&quot;</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="comment">//情况2</span></span><br><span class="line">        <span class="keyword">if</span>(方程.is(<span class="string">&quot;f(x, y&#x27;, y&#x27;&#x27;) = 0 类型&quot;</span>)) &#123;</span><br><span class="line">            System.out.println(<span class="string">&quot;有x, y&#x27;和y&#x27;&#x27;，但是缺y&quot;</span>);</span><br><span class="line">            <span class="keyword">return</span> <span class="string">&quot;第二种可降阶，令y&#x27; = p 降阶，&quot;</span> + </span><br><span class="line">                <span class="string">&quot;会变成一阶线性齐次方程。带公式。如果有必要的话，再降阶一次&quot;</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="comment">//情况3</span></span><br><span class="line">        <span class="keyword">if</span>(方程.is(<span class="string">&quot;f(y, y&#x27;, y&#x27;&#x27;) = 0 类型&quot;</span>)) &#123;</span><br><span class="line">            System.out.println(<span class="string">&quot;有y, y&#x27;和y&#x27;&#x27;，但是缺x（没有单独出现x）&quot;</span>);</span><br><span class="line">            <span class="keyword">return</span> <span class="string">&quot;第三种可降阶，令y&#x27; = p, y&#x27;&#x27; = p*(dp/dy) 降阶，&quot;</span> + </span><br><span class="line">                <span class="string">&quot;会变成一阶线性齐次方程。带公式。如果有必要的话，再降阶一次&quot;</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        </span><br><span class="line">        <span class="comment">//剩下的就是线性的高阶方程</span></span><br><span class="line">        <span class="comment">//对于二阶线性方程，由于更常见，因此单独研究</span></span><br><span class="line">        <span class="comment">//二阶往上的线性方程，原理是类似的，因此没有在算法中列出（三阶常系数线性）</span></span><br><span class="line">        <span class="comment">//二阶线性形如：y&#x27;&#x27; + a(x)*y&#x27; + b(x)*y = c(x)，分为常系数和变系数</span></span><br><span class="line">        <span class="keyword">if</span>(方程.阶数 == <span class="number">2</span>) &#123;</span><br><span class="line">            <span class="comment">//判断是否是线性常系数的</span></span><br><span class="line">            <span class="keyword">if</span>(方程.a(x).equals(<span class="string">&quot;0&quot;</span>) &amp;&amp; 方程.b(x).equals(<span class="string">&quot;0&quot;</span>)) &#123;	<span class="comment">//是线性常系数的</span></span><br><span class="line">                <span class="comment">//判断是否是线性常系数齐次</span></span><br><span class="line">                <span class="keyword">if</span>(方程.c(x).equals(<span class="string">&quot;0&quot;</span>)) &#123;	<span class="comment">//是齐次的</span></span><br><span class="line">                    <span class="comment">//对于齐次线性，只要找到两个线性无关的特解，即可通过公式得到通解</span></span><br><span class="line">                    Expression 二阶特征方程 = 方程.获取对应的特征方程();</span><br><span class="line">                    <span class="keyword">switch</span>(二阶特征方程.根的情况) &#123;	<span class="comment">//可通过求△delta判断</span></span><br><span class="line">                        <span class="keyword">case</span> <span class="string">&quot;λ1 != λ2&quot;</span>:	<span class="comment">//两个不相等的实根</span></span><br><span class="line">                            <span class="keyword">return</span> <span class="string">&quot;带二阶公式：两个不相等的根的通解公式&quot;</span>;</span><br><span class="line">                        <span class="keyword">case</span> <span class="string">&quot;λ1 == λ2&quot;</span>:	<span class="comment">//两个相等的实根</span></span><br><span class="line">                            <span class="keyword">return</span> <span class="string">&quot;带二阶公式：两个相等的根的通解公式&quot;</span>;</span><br><span class="line">                        <span class="keyword">case</span> <span class="string">&quot;λ1, λ2共轭&quot;</span>:	<span class="comment">//两个共轭的根</span></span><br><span class="line">                            <span class="keyword">return</span> <span class="string">&quot;带二阶公式：两个共轭的根的通解公式&quot;</span>;</span><br><span class="line">                    &#125;</span><br><span class="line">                &#125;</span><br><span class="line">                <span class="comment">//剩下的就是非齐次的</span></span><br><span class="line">                <span class="comment">//非齐次的c(x)被人为规定了两种情况:（如果不人为规定，特解非常难找）</span></span><br><span class="line">                <span class="comment">//c(x) = P_n(x)*e^(k*x)或者c(x) = (e^(α*x))*(P(x)*cos(β*x) + Q(x)*sin(β*x))</span></span><br><span class="line">                <span class="comment">//第一种情况</span></span><br><span class="line">                <span class="keyword">if</span>(方程.c(x).equals(<span class="string">&quot;P_n(x)*e^(k*x)&quot;</span>)) &#123;</span><br><span class="line">                    System.out.println(<span class="string">&quot;非齐次型方程的通解 = 对应的齐次的通解 + 非齐次的特解&quot;</span>);</span><br><span class="line">                    StringBuilder s = <span class="keyword">new</span> StringBuilder(<span class="string">&quot;先求对应的齐次的通解，并记录方程的根λ1, λ2&quot;</span>);</span><br><span class="line">                    <span class="comment">//根据多项式P_n(x)的阶数构建新的多项式exp</span></span><br><span class="line">                    <span class="comment">//比如，</span></span><br><span class="line">                    <span class="comment">//P_n(x)的阶数是0，构建的多项式就是a</span></span><br><span class="line">                    <span class="comment">//P_n(x)的阶数是1，构建的多项式就是a*x + b</span></span><br><span class="line">                    <span class="comment">//P_n(x)的阶数是2，构建的多项式就是a*(x^2) + b*x + c</span></span><br><span class="line">                    <span class="comment">//...</span></span><br><span class="line">                    Expression exp = Expression.构建多项式(P_n(x).阶数);</span><br><span class="line">                    <span class="comment">//如果两个根都不等于k</span></span><br><span class="line">                    <span class="keyword">if</span> (c(x).k != λ<span class="number">1</span> &amp;&amp; c(x).k != λ<span class="number">2</span>) &#123;</span><br><span class="line">                        String y_0 = <span class="string">&quot;(exp)*e^(k*x)&quot;</span>;</span><br><span class="line">                        s.append(String.format(</span><br><span class="line">                            <span class="string">&quot;然后令y_0(x) = %s带回，解出a和b，得到非齐次特解，合起来就是通解&quot;</span>, y_0));</span><br><span class="line">                        <span class="keyword">return</span> s.toString();</span><br><span class="line">                    &#125;</span><br><span class="line">                    <span class="comment">//如果两个根都等于k</span></span><br><span class="line">                    <span class="keyword">if</span> (c(x).k == λ<span class="number">1</span> &amp;&amp; c(x).k == λ<span class="number">2</span>) &#123;</span><br><span class="line">                        String y_0 = <span class="string">&quot;(x^2)*(exp)*e^(k*x)&quot;</span>; <span class="comment">//注意最前面多了一个x^2</span></span><br><span class="line">                        s.append(String.format(</span><br><span class="line">                            <span class="string">&quot;然后令y_0(x) = %s带回，解出a和b，得到非齐次特解，合起来就是通解&quot;</span>, y_0));</span><br><span class="line">                        <span class="keyword">return</span> s.toString();</span><br><span class="line">                    &#125;</span><br><span class="line">                    <span class="comment">//剩下的就是两个根只有一个等于k</span></span><br><span class="line">                    String y_0 = <span class="string">&quot;x*(exp)*e^(k*x)&quot;</span>;	<span class="comment">//注意最前面多了一个x</span></span><br><span class="line">                    s.append(String.format(</span><br><span class="line">                        <span class="string">&quot;然后令y_0(x) = %s带回，解出a和b，得到非齐次特解，合起来就是通解&quot;</span>, y_0));</span><br><span class="line">                    <span class="keyword">return</span> s.toString();</span><br><span class="line">                &#125;</span><br><span class="line">                <span class="comment">//第二种情况</span></span><br><span class="line">                <span class="keyword">if</span>(方程.c(x).equals(<span class="string">&quot;(e^(α*x))*(P(x)*cos(β*x) + Q(x)*sin(β*x))&quot;</span>)) &#123;</span><br><span class="line">                    System.out.println(<span class="string">&quot;非齐次型方程的通解 = 对应的齐次的通解 + 非齐次的特解&quot;</span>);</span><br><span class="line">                    StringBuilder s = <span class="keyword">new</span> StringBuilder(<span class="string">&quot;先求对应的齐次的通解，并记录方程的根λ1, λ2&quot;</span>);</span><br><span class="line">                    <span class="comment">//根据多项式P(x)与Q(x)二者中的最高阶数构建新的多项式e1, e2</span></span><br><span class="line">                    <span class="comment">//注意：（1）取二者阶数较高的作为新多项式的阶数。（2）两个多项式的字母不同。</span></span><br><span class="line">                    <span class="comment">//比如，</span></span><br><span class="line">                    <span class="comment">//P(x)的阶数是0，构建的多项式e1就是a，e2就是b</span></span><br><span class="line">                    <span class="comment">//P(x)的阶数是1，构建的多项式e1就是a*x + b，e2就是c*x + d</span></span><br><span class="line">                    <span class="comment">//P(x)的阶数是2，构建的多项式e1就是a*(x^2) + b*x + c，e2就是d*(x^2) + e*x + f</span></span><br><span class="line">                    <span class="comment">//...</span></span><br><span class="line">                    <span class="keyword">int</span> 最高阶数 = Math.max(P(x).阶数, Q(x).阶数);</span><br><span class="line">                    Expression e1 = Expression.构建多项式(最高阶数);	</span><br><span class="line">                    Expression e2 = Expression.构建多项式(最高阶数);</span><br><span class="line">                    <span class="comment">//如果两个根都不等于α + βi</span></span><br><span class="line">                    <span class="keyword">if</span>(α + βi != λ<span class="number">1</span> &amp;&amp; α + βi != λ<span class="number">2</span>) &#123;</span><br><span class="line">                        String y_0 = <span class="string">&quot;(e^(α*x))*((e1)*cos(β*x) + (e2)*sin(β*x))&quot;</span>;</span><br><span class="line">                        s.append(String.format(</span><br><span class="line">                            <span class="string">&quot;然后令y_0(x) = %s带回，解出abcd，得到非齐次特解，合起来就是通解&quot;</span>, y_0));</span><br><span class="line">                        <span class="keyword">return</span> s.toString();</span><br><span class="line">                    &#125;</span><br><span class="line">                    <span class="comment">//如果两个根都等于α + βi</span></span><br><span class="line">                    <span class="keyword">if</span>(α + βi == λ<span class="number">1</span> &amp;&amp; α + βi == λ<span class="number">2</span>) &#123;</span><br><span class="line">                        String y_0 = <span class="string">&quot;(x^2)*(e^(α*x))*((e1)*cos(β*x) + (e2)*sin(β*x))&quot;</span>; <span class="comment">//注意前面多了x^2</span></span><br><span class="line">                        s.append(String.format(</span><br><span class="line">                            <span class="string">&quot;然后令y_0(x) = %s带回，解出abcd，得到非齐次特解，合起来就是通解&quot;</span>, y_0));</span><br><span class="line">                        <span class="keyword">return</span> s.toString();</span><br><span class="line">                    &#125;</span><br><span class="line">                    <span class="comment">//剩下的就是两个根只有一个等于α + βi</span></span><br><span class="line">                    String y_0 = <span class="string">&quot;x*(e^(α*x))*((e1)*cos(β*x) + (e2)*sin(β*x))&quot;</span>;	<span class="comment">//注意最前面多了一个x</span></span><br><span class="line">                    s.append(String.format(</span><br><span class="line">                        <span class="string">&quot;然后令y_0(x) = %s带回，解出abcd，得到非齐次特解，合起来就是通解&quot;</span>, y_0));</span><br><span class="line">                    <span class="keyword">return</span> s.toString();</span><br><span class="line">                &#125;</span><br><span class="line">            &#125;</span><br><span class="line">            <span class="comment">//剩下的就是线性变系数的：也分为齐次和非齐次</span></span><br><span class="line">            <span class="comment">//不需要会解，只要能明白解的结构即可</span></span><br><span class="line">            <span class="comment">//齐次：</span></span><br><span class="line">            <span class="keyword">if</span>(方程.c(x).equals(<span class="string">&quot;0&quot;</span>)) &#123;</span><br><span class="line">                System.out.println(<span class="string">&quot;通解公式:y = c_1*φ_1(x) + c_2*φ_2(x)&quot;</span>);</span><br><span class="line">                <span class="keyword">return</span> String.format(</span><br><span class="line">                    <span class="string">&quot;高阶线性变系数齐次方程，找到%d个线性无关的特解，带公式得到通解&quot;</span>, 方程.阶数);</span><br><span class="line">            &#125;</span><br><span class="line">			<span class="comment">//非齐次：除了齐次就是非齐次</span></span><br><span class="line">            System.out.println(<span class="string">&quot;非齐次型方程的通解 = 对应的齐次的通解 + 非齐次的特解&quot;</span>);</span><br><span class="line">            <span class="keyword">return</span> <span class="string">&quot;高阶线性变系数非齐次方程，找到对应的齐次的特解，通过通解公式得到齐次的通解&quot;</span> + </span><br><span class="line">                <span class="string">&quot;然后加入非齐次的特解，合起来就是非齐次通解&quot;</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="comment">//二阶往上的方程，由于原理相同，且做题时比较少见，不再列出。</span></span><br><span class="line">        <span class="comment">//（仅列出三阶线性常系数）</span></span><br><span class="line">        <span class="keyword">if</span>(方程.阶数 == <span class="number">3</span>) &#123;</span><br><span class="line">            <span class="keyword">if</span>(方程.a(x).equals(<span class="string">&quot;0&quot;</span>) &amp;&amp; 方程.b(x).equals(<span class="string">&quot;0&quot;</span>)) &#123;	<span class="comment">//是线性常系数的</span></span><br><span class="line">                <span class="comment">//判断是否是齐次的</span></span><br><span class="line">                <span class="keyword">if</span>(方程.c(x).equals(<span class="string">&quot;0&quot;</span>)) &#123;</span><br><span class="line">					Expression 三阶特征方程 = 方程.获取对应的特征方程();</span><br><span class="line">                      <span class="keyword">switch</span>(三阶特征方程.根的情况) &#123;	</span><br><span class="line">                          <span class="keyword">case</span> <span class="string">&quot;λ1, λ2, λ3都是实数且λ1 != λ2 != λ3&quot;</span>:</span><br><span class="line">                              <span class="keyword">return</span> <span class="string">&quot;带三阶公式：对应二阶的case 2&quot;</span>;</span><br><span class="line">                          <span class="keyword">case</span> <span class="string">&quot;λ1, λ2, λ3都是实数，且λ1 = λ2 != λ3&quot;</span>:</span><br><span class="line">                              <span class="keyword">return</span> <span class="string">&quot;带三阶公式。没有对应的二阶&quot;</span>;</span><br><span class="line">                          <span class="keyword">case</span> <span class="string">&quot;λ1, λ2, λ3都是实数，且λ1 = λ2 = λ3&quot;</span>:</span><br><span class="line">                              <span class="keyword">return</span> <span class="string">&quot;带三阶公式：对应二阶的case 1&quot;</span>;</span><br><span class="line">                          <span class="keyword">case</span> <span class="string">&quot;λ1是实数, λ2与λ3共轭&quot;</span>:</span><br><span class="line">                              <span class="keyword">return</span> <span class="string">&quot;带三阶公式，没有对应的二阶&quot;</span>;</span><br><span class="line">                     &#125;</span><br><span class="line">                &#125;</span><br><span class="line">                <span class="comment">//剩下的就是非齐次的（老师没讲）</span></span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">	&#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="class"><span class="keyword">interface</span> 常数值 </span>&#123;</span><br><span class="line">    <span class="comment">//定义常用的常数值</span></span><br><span class="line">	<span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">final</span> <span class="keyword">int</span> ERROR = Integer.MIN_VALUE;</span><br><span class="line">    <span class="comment">//省略代码</span></span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">Expression</span> <span class="keyword">implements</span> 常数值</span>&#123;</span><br><span class="line">    <span class="comment">//省略代码</span></span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> 微分方程 <span class="keyword">implements</span> 常数值 </span>&#123;</span><br><span class="line">	<span class="keyword">public</span> Expression 表达式;</span><br><span class="line">    <span class="keyword">public</span> <span class="keyword">int</span> 阶数;</span><br><span class="line">    <span class="function"><span class="keyword">public</span> Expression <span class="title">a</span><span class="params">(x)</span> </span>= ERROR; <span class="comment">//a(x)不一定存在（只有线性的方程存在），因此预先定义为ERROR</span></span><br><span class="line">	<span class="function"><span class="keyword">public</span> Expression <span class="title">b</span><span class="params">(x)</span> </span>= ERROR;	<span class="comment">//b(x)不一定存在（只有线性的方程存在），因此预先定义为ERROR</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> Expression <span class="title">c</span><span class="params">(x)</span> </span>= ERROR;	<span class="comment">//c(x)不一定存在（只有线性的方程存在），因此预先定义为ERROR</span></span><br><span class="line">    </span><br><span class="line">    <span class="keyword">public</span> 微分方程(Expression 表达式) &#123;</span><br><span class="line">        <span class="keyword">this</span>.表达式 = 表达式;</span><br><span class="line">        <span class="keyword">this</span>.阶数 = 表达式.最高阶导数的阶数;</span><br><span class="line">        <span class="keyword">this</span>.a(x) = 表达式.a(x);</span><br><span class="line">        <span class="keyword">this</span>.b(x) = 表达式.b(x);</span><br><span class="line">        <span class="keyword">this</span>.c(x) = 表达式.c(x);</span><br><span class="line">    &#125;</span><br><span class="line">	<span class="comment">/**</span></span><br><span class="line"><span class="comment">	 * 判断微分方程是否满足特定要求</span></span><br><span class="line"><span class="comment">	 * <span class="doctag">@param</span> s: 判断条件</span></span><br><span class="line"><span class="comment">	 */</span></span><br><span class="line">	<span class="function"><span class="keyword">public</span> <span class="keyword">boolean</span> <span class="title">is</span><span class="params">(String s)</span> </span>&#123;</span><br><span class="line">		<span class="comment">//省略代码</span></span><br><span class="line">	&#125;</span><br><span class="line">    <span class="comment">/**</span></span><br><span class="line"><span class="comment">     * 得到微分方程的特征方程</span></span><br><span class="line"><span class="comment">     */</span></span><br><span class="line">    <span class="keyword">public</span> Expression 获取对应的特征方程() &#123;</span><br><span class="line">        <span class="comment">//省略代码</span></span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">Expression</span> </span>&#123;</span><br><span class="line">    <span class="comment">//省略代码</span></span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

    </div>

    
    
    

    <footer class="post-footer">
          <div class="reward-container">
  <div></div>
  <button onclick="document.querySelector('.post-reward').classList.toggle('active');">
    赞赏
  </button>
  <div class="post-reward">
      <div>
        <img src="https://cdn.jsdelivr.net/gh/ysl970629/public_picture_bed_01@latest/img/2020-04-21 210252.jpg" alt="YuSLi 微信">
        <span>微信</span>
      </div>
      <div>
        <img src="https://cdn.jsdelivr.net/gh/ysl970629/public_picture_bed_01@latest/img/2020-04-21 210252(1).jpg" alt="YuSLi 支付宝">
        <span>支付宝</span>
      </div>

  </div>
</div>

          <div class="post-tags">
              <a href="/tagsa/%E6%95%B0%E5%AD%A6%E4%BA%8C/" rel="tag"># 数学二</a>
              <a href="/tagsa/%E9%AB%98%E6%95%B0/" rel="tag"># 高数</a>
              <a href="/tagsa/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B/" rel="tag"># 微分方程</a>
          </div>

        

          <div class="post-nav">
            <div class="post-nav-item">
                <a href="/posts/3157ddc2/" rel="prev" title="常用的文章链接">
                  <i class="fa fa-chevron-left"></i> 常用的文章链接
                </a>
            </div>
            <div class="post-nav-item">
                <a href="/posts/38420ac9/" rel="next" title="Windows下关闭指定端口号的进程">
                  Windows下关闭指定端口号的进程 <i class="fa fa-chevron-right"></i>
                </a>
            </div>
          </div>
    </footer>
  </article>
</div>






    <div class="comments" id="valine-comments"></div>

<script>
  window.addEventListener('tabs:register', () => {
    let { activeClass } = CONFIG.comments;
    if (CONFIG.comments.storage) {
      activeClass = localStorage.getItem('comments_active') || activeClass;
    }
    if (activeClass) {
      const activeTab = document.querySelector(`a[href="#comment-${activeClass}"]`);
      if (activeTab) {
        activeTab.click();
      }
    }
  });
  if (CONFIG.comments.storage) {
    window.addEventListener('tabs:click', event => {
      if (!event.target.matches('.tabs-comment .tab-content .tab-pane')) return;
      const commentClass = event.target.classList[1];
      localStorage.setItem('comments_active', commentClass);
    });
  }
</script>
</div>
  </main>

  <footer class="footer">
    <div class="footer-inner">


<div class="copyright">
  &copy; 
  <span itemprop="copyrightYear">2020</span>
  <span class="with-love">
    <i class="fa fa-heart"></i>
  </span>
  <span class="author" itemprop="copyrightHolder">YuSLi</span>
</div>
<div class="wordcount">
  <span class="post-meta-item">
    <span class="post-meta-item-icon">
      <i class="fa fa-chart-line"></i>
    </span>
    <span title="站点总字数">11k</span>
  </span>
  <span class="post-meta-item">
    <span class="post-meta-item-icon">
      <i class="fa fa-coffee"></i>
    </span>
    <span title="站点阅读时长">18 分钟</span>
  </span>
</div>
  <div class="powered-by">由 <a href="https://hexo.io/" class="theme-link" rel="noopener" target="_blank">Hexo</a> & <a href="https://theme-next.js.org/" class="theme-link" rel="noopener" target="_blank">NexT.Gemini</a> 强力驱动
  </div>

    </div>
  </footer>

  
  <script src="//cdn.jsdelivr.net/npm/animejs@3.2.1/lib/anime.min.js"></script>
<script src="/js/utils.js"></script><script src="/js/motion.js"></script><script src="/js/next-boot.js"></script><script src="/js/bookmark.js"></script>

  
  <script>
    (function(){
      var bp = document.createElement('script');
      var curProtocol = window.location.protocol.split(':')[0];
      bp.src = (curProtocol === 'https') ? 'https://zz.bdstatic.com/linksubmit/push.js' : 'http://push.zhanzhang.baidu.com/push.js';
      var s = document.getElementsByTagName("script")[0];
      s.parentNode.insertBefore(bp, s);
    })();
  </script>

<script src="/js/local-search.js"></script>






  





  <script>
  if (typeof MathJax === 'undefined') {
    window.MathJax = {
      tex: {
        inlineMath: {'[+]': [['$', '$']]},
        tags: 'ams'
      },
      options: {
        renderActions: {
          findScript: [10, doc => {
            document.querySelectorAll('script[type^="math/tex"]').forEach(node => {
              const display = !!node.type.match(/; *mode=display/);
              const math = new doc.options.MathItem(node.textContent, doc.inputJax[0], display);
              const text = document.createTextNode('');
              node.parentNode.replaceChild(text, node);
              math.start = {node: text, delim: '', n: 0};
              math.end = {node: text, delim: '', n: 0};
              doc.math.push(math);
            });
          }, '', false],
          insertedScript: [200, () => {
            document.querySelectorAll('mjx-container').forEach(node => {
              const target = node.parentNode;
              if (target.nodeName.toLowerCase() === 'li') {
                target.parentNode.classList.add('has-jax');
              }
            });
          }, '', false]
        }
      }
    };
    const script = document.createElement('script');
    script.src = '//cdn.jsdelivr.net/npm/mathjax@3.1.2/es5/tex-mml-chtml.js';
    script.defer = true;
    document.head.appendChild(script);
  } else {
    MathJax.startup.document.state(0);
    MathJax.typesetClear();
    MathJax.texReset();
    MathJax.typeset();
  }
</script>



<script>
NexT.utils.loadComments('#valine-comments', () => {
  NexT.utils.getScript('//cdn.jsdelivr.net/npm/valine@1.4.14/dist/Valine.min.js', () => {
    new Valine(Object.assign({"enable":true,"appId":"f8FLjTWsKkdxHhmlhnHyvK6o-MdYXbMMI","appKey":"0aa9z8gRXlm6cczqYR1WQBxj","notify":false,"verify":false,"placeholder":"写下你的评论...","avatar":"mm","meta":["nick","mail","link"],"pageSize":10,"visitor":false,"comment_count":true,"recordIP":false,"serverURLs":null}, {
      el: '#valine-comments',
      path: "/posts/d6882305/",
      serverURLs: "https://f8fljtws.api.lncldglobal.com"
    }));
  }, window.Valine);
});
</script>

</body>
</html>
